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Question Number 175588 by alcohol last updated on 03/Sep/22

Commented by mahdipoor last updated on 04/Oct/22

$$210=2\times 3\times 5\times 7$$$$in2022!=2^a\times 3^b\times 5^c\times 7^d\times ...$$$$a⩾b⩾c⩾d⩾...\Rightarrow$$$$max\left(n\right)if\frac{2022!}{{210}^n}\in N=max\left(d\right)if\frac{2022!}{7^d}\in N$$$$d=\left[\frac{2022}{7}\right]+\left[\frac{2022}{7^2}\right]+\left[\frac{2022}{7^3}\right]+\left[\frac{2022}{7^4}\right]+...=$$$$288+41+5+0+...=334$$

Answered by BaliramKumar last updated on 13/Mar/23

$$$$$$210=2\times 3\times 5\times 7$$$$$$$$\begin{array}{|cc|}\hline 7& 2022\\ 7& 288\\ 7& 41\\ 7& 5\\ & 0\\ \hline\end{array}or\overline{)\begin{array}{c}\lfloor \frac{2022}{7}\rfloor =288\\ \lfloor \frac{288}{7}\rfloor =41\\ \lfloor \frac{41}{7}\rfloor =5\end{array}}$$$$288+41+5=334$$

Commented by Tawa11 last updated on 15/Sep/22

$$\mathrm{Great}\mathrm{sir}$$

Answered by mr W last updated on 03/Sep/22

$$\frac{2022!}{{210}^n}=k$$$$2022!={210}^{n}k$$$$2022!=2^n{3}^n{5}^n{7}^{n}k$$$$2022!=2^{2014}\times 3^{1006}\times 5^{503}\times 7^{334}\times ...$$$$\Rightarrow n_{max}=334$$$$seealsoQ115294$$

Commented by Tawa11 last updated on 15/Sep/22

$$\mathrm{Great}\mathrm{sir}$$

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